Stochasticity driving robust pattern formation in brain wiring

During brain development, synaptic connection patterns are formed in an extremely robust manner. As the interconnection patterns are much too complex to be encoded directly in the genome, they must emerge from simpler rules. In this project we investigate mechanistic stochastic models of axon growth and filopodial dynamics, checking whether their simulation leads to connection patterns and dynamics as observed in vivo, and with the same robustness.

During the development of the brain, the neurons form specific patterns in a very robust andreliable way. The question of how the axons and dendrites find the appropriate synaptic partnershas been studied for decades, but is posed today with a new twist.

Axon guidance revisited

For the last 70 years, the dominant model of axon guidance due to R. Sperry have been globalconcentration gradients of guidance molecules, of which about 100 different species are knowntoday. During axon growth -- filopodia, small extrusions of the growth cone -- sprout in differentdirections, as has been observed in histological microscopy images. The general assumptionhas been that the filopodia sample the surroundings for the chemical gradients of guidancemolecules, allowing to find the right direction despite the stochasticity of filopodia growth andmolecule sensing.

With the new technology of multi-photon time-lapse microscopy, the biologist R. Hiesinger (FU)is able to acquire in vivo 4D microscopy movies of axons growing in drosophila brains -- andsheds new light on axon guidance and the brain wiring process. The filopodial dynamics featuresa much richer structure than would be necessary for stochastic gradient sampling (Figure 1).Currently, the role that this complex dynamics plays in the brain wiring process is essentiallyunknown.

Looking for simplicity

The drosophila genome is too small for encoding the brain wiring explicitly. Thus, the patternmust emerge from rather simple regulatory mechanisms that are encoded in the genome [1].One compelling hypothesis is that these developmental rules do not only tolerate randomness inthe axons’ environment, but use stochasticity as a driving force and to achieve robustness.In a joint MATHEON project with M. von Kleist (FU), we aim at identifying mechanistic modelsthat are both physically plausible and able to reproduce observed wiring patterns and thestatistics of filopodial dynamics (Figure 2). The simpler and more general the mechanismsforming such a model can be, the more one can expect that structurally similar processes areactually driving the neural development.

Figure 2: Statistics of filopodia dynamics extracted from 4D microscopy data. Characteristic quantities such as filopodia length (histogram of lenghts taken from an hour of observation at P+40% and P+60%) filopodia extension and retraction events follow a Poisson distribution.

As a first candidate, we consider a model comprising three essential components. First,diffusion and decay of guidance molecules in the extracellular space seems necessary for inter-axon communication. Reception of guidance molecules by the filopodia is probably a stochasticprocess due to the low concentration. And finally, nonlinear reactions going on within the axons,and affected by the sensing of guidance molecules, control growth and retraction of filopodia aswell as eventually the release of guidance molecules [2]. Intracellular diffusion and directedtransport may also play an essential role.

For simulating realizations of such models, solvers for deterministic partial differential equationsdescribing diffusion, reaction, and transport processes need to be coupled to a stochasticsimulation algorithm capturing the random events of guidance molecule reception and filopodiagrowth.

Model complexity can be measured in terms of the number of involved species of guidancemolecules, and the number of nonlinear reactions. One of the simplest models of this type,containing just a single type of guidance molecule, can already create robust and quasi-regularspace-filling axon structures that avoid self-contact as well as neighbour contact.

For the stochasticity in the model, the statistics for length, lifetime, extension and retraction eventsas well as birth and death rate such as in Figure 2 are evaluated.They are used in a Gillespie or chemical master equation algorithm, to test and find the transition propensities for a filopodium to become a bulbous tip on synapse partner contact (reversible) andfor a bulbous tip to become a stable synapse (irreversible), cf. Figure 4.Irrespective of the bulbous tip forming a synapse or forming back to a filopodium, on becoming abulbous tip, a feedback to the axon's growth cone takes place, that inhibits the filopodial activity.This regulates observably the number of synapses being formed.

It has also been found, that should molecules 'DLar' or 'Liprin-a' (read alpha) be missing, thefeedforward from bulbous tips to become synapses (Figure 4, green arrow) is inhibited renderingless synapses. Should the molecule 'trio' be missing, the feedback to filopodial dynamic reductionis flawed, which leads to more observed bulbous tips than in the wild type axon. The molecule'Syd1' has an impact on both feeds.

The effect of these molecules is also to be included and tested in the model. Found transitionpropensities will then be transferred to the geometric model (Figure 3) and cross checked tofind what role spatial distribution plays or how synapse partners need to be spatially distributedin order to achieve the same propensities.

Figure 4: Transition model from filopodium to synapse. On filopodial contact so-called bulbous tipsare formed, on which the filopodial dynamics is damped (red feedback). The bulbous tips can eitherstabilize and become synapses or transition back to a filopodium. Novel model developed byR. Hiesinger et. al.